Grain sorting and bar instability - unipd.it · PDF fileGrain sorting and bar instability By STEFANO LANZONI1 AND MARCO TUBINO2 1Dipartimento di Ingegneria Idraulica, ... the bed topography

J. Fluid Mech. (1999), vol. 393, pp. 149–174. Printed in the United Kingdom

c© 1999 Cambridge University Press

149

Grain sorting and bar instability

By S T E F A N O L A N Z O N I1 AND M A R C O T U B I N O2

1Dipartimento di Ingegneria Idraulica, Marittima e Geotecnica, Universita di Padova,Via Loredan 20, 35131 Padova, Italy

e-mail: [email protected] di Ingegneria Civile e Ambientale, Universita di Trento,

Via Mesiano 77, 38100 Trento, Italye-mail: [email protected]

(Received 5 January 1998 and in revised form 15 January 1999)

A two-dimensional model of flow and bed topography is proposed to investigatethe effect of sediment heterogeneity on the development of alternate bars. Within thecontext of a linear stability theory the flow field, the bed topography and the grain sizedistribution function are perturbed leading to an integro-differential linear eigenvalueproblem. It is shown that the selective transport of different grain size fractions andthe resulting spatial pattern of sorting may appreciably affect the balance betweenstabilizing and destabilizing actions which govern bar instability. Theoretical resultssuggest that sediment heterogeneity leads to a damping of both growth rate andmigration speed of bars, while bar wavelength is shortened with respect to thecase of uniform sediment. The above findings conform, at least qualitatively, to theexperimentally detected reduction of bar height, length and celerity. The observedtendency of coarser particles to pile up towards bar crests is also reproduced bytheoretical results.

1. IntroductionAlternate bars are the most common type of large-scale bedforms which are

observed both in sandy streams and in gravel bed rivers. They essentially consist ofmigrating alternating regions of scour and deposit with horizontal scale of the orderof several channel widths and vertical scale of the order of the flow depth. Theirvertical spatial scale being so large, the problem of predicting their occurrence andtheir equilibrium characteristics has a profound impact on several aspects of riverengineering.

A fairly consistent picture of the process underlying bar development in cohesionlesschannels has been built up through a large number of theoretical and experimentalworks, which have been mainly developed in the last two decades. In particular, ithas been progressively established that the occurrence of bars and their response tothe planimetric evolution of channels are crucial ingredients required to characterizethe overall morphology of alluvial channels at the river reach scale (see the state ofthe art reviews of Seminara & Tubino 1989, and Seminara 1995).

The formation of river bars has been conclusively explained in terms of an in-herent instability of an erodible bed subject to a turbulent flow in straight channel,which enables the growth of infinitesimal flow and bottom perturbations migratingdownstream. Several increasingly refined two-dimensional linear stability theories areavailable in the literature for the case when the dominant form of sediment transport

150 S. Lanzoni and M. Tubino

is bedload (see e.g. Callander 1969; Parker 1976; Fredsoe 1978; Colombini, Seminara& Tubino 1987; Garcia & Nino 1993). These theories allow one to predict the growthrate of perturbations within the linear regime, the marginal stability conditions in thespace of flow and sediment parameters and the wavelength and wave speed selectedby the instability mechanism, i.e. those corresponding to the maximum growth rate.In particular, it appears that in bedload dominated channels the width to depth ratioβ is the control parameter of bar instability such that a threshold value βc can bedetermined, for given Shields parameter Θ and relative grain roughness ds, abovewhich bars are expected to grow. The balance between stabilizing and destabilizingactions yielded by linear theory also suggests that the alternate bar mode is the mostunstable, the higher modes (leading for example to central or multiple bar config-urations) prevailing only for values of β which are much larger than the thresholdvalue.

Furthermore, Colombini et al. (1987) and Colombini & Tubino (1991) have inves-tigated nonlinear interactions of various modes arising when β exceeds the thresholdvalue, showing that nonlinear effects cause bed perturbations to reach an equilibriumfinite amplitude characterized by periodic diagonal fronts. Under suitable conditions,which are rarely encountered in nature, the above equilibrium solution may in turnbecome unstable and bifurcate into a more complex quasi-periodic pattern (Schielen,Doelman & De Swart 1993).

Whether the width-to-depth ratio keeps the role of control parameter of barinstability when suspended load prevails, i.e. in sandy streams, is still an openquestion. Suspended load has been found to be invariably destabilizing, i.e. enhancingbar development, by Fredsoe (1978) and Kuroki (1988); on the other hand, Watanabe& Tubino’s (1992) results suggest that for given grain size a threshold value of Shieldsstress (i.e. of suspended load intensity) exists above which bar perturbations aresuppressed, the critical value βc becoming infinite. Furthermore, the three-dimensionalanalysis of Repetto, Tubino & Zolezzi (1996) introduces a more subtle mechanismof suppression of alternate bars in that it is found that suspension enhances theoccurrence of higher transverse modes which may then take over even for relativelylow values of β.

The analyses on river bars mentioned above have concentrated on the case ofuniform sediment. This is partly motivated by the complexity and cost of laboratorywork on sediment mixtures, especially when dealing with large-scale features whichinvolve a considerable amount of sediments. Furthermore, significant progress inobtaining knowledge of the physical mechanisms underlying bedload transport ofsediment mixtures has been achieved only quite recently (see the extensive review ofParker 1992).

Nevertheless, the mixed character of the sediment is a significant characteristicof river beds, particularly of gravel bed rivers. A wide range of features of gravelrivers cannot be adequately explained without reference to the entire particle sizedistribution as well as to the related morphological characteristics of gravel beds(armouring, vertical sorting). The selective transport of individual size fractions ofa mixture, which is induced by bed deformation and spatial non-uniformity of bedshear stress distribution, results in a consistent pattern of longitudinal and transversesorting which in turn affects the balance between stabilizing and destabilizing effectscontrolling bedform development. This is suggested by several field and flume data(Lisle, Ikeda & Iseya 1991; Diplas & Parker 1992; Lisle & Madej 1992; Ashworth,Ferguson & Powell 1992b; Lanzoni 1996). Furthermore, field observations of Whitinget al. (1985) and theoretical results of Seminara, Colombini & Parker (1996) suggest

Grain sorting and bar instability 151

that there exists a class of morphological patterns (called bedload sheets) which areinherently associated with the heterogeneous character of the sediment and may be thecause of temporal and spatial variations of bedload transport typical of gravel rivers.

The above considerations strongly motivate the need to investigate the role ofsediment heterogeneity on bar instability. The basic questions to be answered are:How does the heterogeneous character of sediment affect the mechanism which givesrise to bars in the uniform case? What is the spatial distribution of different grainsizes induced by bar development? Does sediment non-uniformity introduce in thestability analysis further ingredients which are inherently associated with particlesize distribution? Indeed, the spatial variations of both bottom roughness, which areinduced by sorting, and of bedload transport, which are triggered by the deviationfrom equal mobility of different sizes, constitute additional mechanisms able to modifythe instability process with respect to the case of uniform sediment.

From the technical point of view the fundamental question posed by the mixedcharacter of sediments is that of ascertaining whether the region of bar occurrenceand the depth of scour and depositional regions change when sediment heterogeneityis taken into account. Experimental findings seem to suggest that the topography ofbars is significantly modified. In particular, sediment non-uniformity has been foundto induce a somewhat episodic bar growth, even under steady hydraulic conditions,and to prevent downstream migration of bars by inhibiting sediment transport overthem (Lisle et al. 1991). Moreover, depending on the average Shields stress andrelative bed roughness, a significant reduction of bar amplitude and change of barwavelength have been detected compared to the case of uniform sediment (Lanzoni,Tubino & Bruno 1994; Lanzoni 1996).

In the present investigation a suitable treatment of sediment mixtures, whereby themodel of Parker (1990) is adopted along with Hirano’s (1971) concept of active layer,is incorporated within the framework of a linear stability analysis of flow and bottomtopography. Since sediment heterogeneity mainly characterizes gravel bed rivers, weneglect suspended load. Therefore a simple two-dimensional description of flow andsediment transport is adopted.

A similar stability analysis for the grain size distribution has been recently proposedby Seminara et al. (1996) to explain the formation of bedload sheets. The mostnoticeable difference introduced by the present work is that the full coupling betweenperturbations of bottom topography and grain size distribution is retained.

The main output of the theory is a significant damping of bar instability, which isimplied by the decrease of the growth rate of bar perturbations for increasing valuesof the standard deviation of sediment mixture. On the other hand, the critical valueof width ratio for the occurrence of bars does not seem to be significantly affected bythe heterogeneous character of the sediment, the values of βc falling slightly aboveor below those obtained in the uniform case depending on the dimensionless grainroughness.

Furthermore, theoretical results suggest that the pattern of sorting which is in-duced by the development of bottom topography crucially depends on the relativeimportance of the mechanisms which are responsible for the spatial displacement ofparticles. In particular, while gravitational effects tend to pull selectively coarser par-ticles downward, the unequal response of grain sizes to spatial variations of bottomstress seems to promote the piling up of coarser particles towards bar crests. Whenthe spatial variations of bottom topography are relatively slow, as in the case of riverbars, the latter mechanism seems to prevail leading to the progressive coarseningwhich is often observed along the upstream face of bars.


Further related questions regarding the effect of grain sorting on the equilibriumamplitude of bars will require additional investigation where nonlinear effects areaccounted for.

The rest of the paper is organized as follows. In § 2 we formulate the problem. Thelinear stability analysis is reported in § 3, while in § 4 we derive an analytical solutionof the dispersion relationship arising from linear theory for the case when the averagegrain size distribution function is represented in terms of Dirac distributions. Finally,§ 5 contains a discussion of the results along with some concluding remarks.

2. Formulation of the problemLet us consider the flow in a straight alluvial channel with non-erodible banks and

constant width 2B∗ large enough for the flow to be modelled as two-dimensional. Thechannel bottom is assumed to be made up of a mixture of cohesionless sedimentsof density ρs. The probability density function describing the distribution of grainsizes available for bedload transport which are contained within the surface layer isdenoted by f(φ, s∗, n∗, t∗). Here φ is the sedimentological scale for particle size definedsuch that the grain diameter (in mm) is d∗ = 2−φ, s∗ and n∗ are the longitudinaland transverse Cartesian coordinates and t∗ is time (here and in the following a starsuperscript denotes dimensional quantities).

The first preliminary question which needs to be addressed is that of selecting asuitable model for the probability density function f. Plots of cumulative frequenciesof grain size distributions of sediments transported in nature versus the logarithms ofgrain sizes are generally line segments. A common explanation of this behaviour isthat these segments result from log-normal subpopulations that are either truncated oroverlapped (Visher 1969; Tanner 1983), each transported by a particular mechanism(bedload, saltation, suspended load). Nevertheless, sediments transported by a singlemechanism may also display segmented patterns since a prolonged transportationas bedload or suspended load is required to let the grain size distribution becomenearly perfectly log-normal (Sengupta, Ghosh & Mazumder 1991). Quite often grainsize distributions of natural sediments are found to fit a log-hyperbolic distribution(Bagnold & Barndorff-Nielsen 1980) or a bimodal distribution (Shaw & Kellerhals1982). Estimation of the four parameters needed for the specification of a hyperbolicdistribution, however, is rather complicated as discussed in Christiansen & Hartman(1991). Therefore, in the following we will assume that the gross features of the grainsize distribution can be adequately described by the first and second moments of thedistribution. Hence we can write

φg =

∫ ∞−∞fφ dφ, σ2 =

∫ ∞−∞

(φ− φg)2f dφ, (2.1a,b)

which allow us to define the geometric mean grain diameter d∗g = 2−φg and its standarddeviation σg = 2σ , respectively. By definition the probability density function is subjectto the following integral condition:∫ ∞

−∞f(φ, s∗, n∗, t∗) dφ = 1. (2.2)

It is worth pointing out that the use of a probability density function f which isindependent of the vertical coordinate rules out the possibility of taking into accountthe effect of vertical sorting on bar development. The above assumption, which goesback to the original formulation of the Exner equation for size fractions introduced


2B*

n*

s*

Bar crest(deposition)

Pool(erosion)

Bar front

Lb*

Referenceplane

Channelbed

Watersurface

s*

D*

H *n*

V *

U *

2B *

Figure 1. Sketch of the channel and notation.

by Hirano (1971) and has been successfully employed by Seminara et al. (1996) toexplain the formation of gravel sheets, may only be justified in the case of bedformswith negligible vertical scale, while it may result into a too crude approximation whenapplied in a context in which bedform development and migration act significantly tomix the sediment within the bed layer. The case of dunes in heterogeneous sedimentsis a notable example where vertical sorting may play a significant role (see Ribberink1987). Some recent attempts to characterize vertical sorting through a continuousvertical size distribution are documented in the literature (see e.g. Armanini 1995).However, since we are concerned with the process of incipient formation of bars wemay discard this effect, which greatly simplifies the theoretical treatment. Moreover,unlike in the dune case, here the relevant spatial scale of flow and bottom non-uniformity is channel width which is typically much larger than flow depth.

We wish to analyse how the flow field responds to simultaneous two-dimensionalinfinitesimal perturbations both of the grain size distribution and of the bottomconfiguration, assuming that the fluid can adapt instantaneously to such perturbations.Our basic state is a uniform flow over a flat cohesionless bed, with flow depth D∗0and average speed U∗0 , characterized by a grain size probability density functionf0(φ). Let d∗g0 denote the geometric mean grain size of such a distribution, which isuniformly distributed in space. The effect of lateral distribution of grain size, suchas that associated with the presence of discrete patches (Paola & Seal 1995), can beaccounted for by setting a suitable initial distribution. Also, note that the influenceof local composition on grain mobility is felt through the dependence of the hidingfunction on local mean grain diameter (see equation (2.15) below).

The formulation of the problem for the flow follows closely the work of Colombini etal. (1987) and will only briefly be recalled here. A steady two-dimensional formulationis adopted in as much as spatial variations occur on planimetric scales much largerthan the flow depth and bedforms develop quite slowly. Sidewall layers are ignored andreplaced by the condition of vanishing transverse component of the depth-averagedvelocity.

The governing equations for the flow then read

UU,s + VU,n = −H,s − β τs/D, (2.3a)

UV,s + VV,n = −H,n − β τn/D, (2.3b)

(UD),s + (VD),n = 0, (2.3c)

where (U, V ) is the depth-averaged velocity vector, H is the free-surface elevation,


(τs, τn) are longitudinal and transverse components of bottom shear stress and D isthe local flow depth (see figure 1). The variables have been made dimensionless in theform

(U∗, V ∗) = U∗0 (U,V ), (D∗, H∗) = D∗0(D, F2r H), (2.4a,b)

(s∗, n∗) = B∗(s, n), (τ∗s , τ∗n) = ρU∗20 (τs, τn), (2.4c,d )

where ρ is fluid density, Fr is the Froude number of uniform basic flow, and the widthratio β is:

β = B∗/D∗0 . (2.5)

Closure of (2.3a, c) requires a suitable constitutive relationship for bottom stress.We thus express the shear stress in terms of a friction coefficient C defined by therelationship

(τs, τn) = C(U,V )(U2 + V 2)1/2. (2.6)

The structure of C strongly depends on the presence of small-scale (ripples) and/ormesoscale (dunes) bedforms which are often found to coexist with alternate bars,particularly in sandy rivers. Here we restrict attention to gravel bed rivers in which,although the bottom may contain a certain amount of sand, bedload is dominant anddunes are less important. Incidentally, note that some experimental evidence existswhich suggests that dune height tends to be damped due to sorting effects (Klaassen1990; Lanzoni 1996).

Assuming local equilibrium we can express C in terms of the local roughness usingthe following relationship:

C−1/2 = 6 + 2.5 ln (D/ks), (2.7)

which strictly applies to uniform conditions and a plane bed, where the dimensionlessgrain roughness height (scaled by D∗0)

ks = nσdσ (2.8)

is taken to be proportional to a coarse grain size defined as d∗σ = d∗g2σ , the constantof proportionality nσ being roughly equal to 2 (Parker 1992).

It is worth pointing out that through the above definition of local roughness theheterogeneous character of the sediment introduces a first novel feature in the stabilityproblem in that perturbations of the (local) average grain size, which are inducedby sorting effects associated with bar development, produce through equation (2.6)perturbations of bottom stress. Whether the above effect is able to enhance or dampbar instability can only be ascertained through the determination of the sortingpattern associated with bar topography. We will see that in the case of bars theprogressive coarsening observed along the upstream face of bedforms produces alocal increase of roughness there, i.e. of shear stress, whose effect cumulates with theperturbed distribution of longitudinal stress typical of bars with uniform sediment.

Finally, we require that the sidewalls must be impermeable to the fluid; hence wewrite

V = 0 (n = ± 1). (2.9)

The mathematical problem governing the motion of the fluid is then coupled to theconservation equations for the sediment. The formulation of the problem of transportand mass balance of each size fraction within the mixture follows closely that ofSeminara et al. (1996). Therefore we will simply summarize it (see also Parker 1992).


We assume that sediments available for transport are contained in a surface-activelayer (Hirano 1971) which is taken to be initially well mixed so that the probabilitydensity function f of the material within it has neither a vertical structure nor alongitudinal or a transverse structure.

The governing equations for the motion of the sediment mixture are: a statementof mass balance for each size fraction, which allows for variation of bottom elevationin time and transverse variations of bedload transport; a dynamic equation to specifythe bedload transport rate of each fraction. The former reads

fη,t + Laf,t + [(fQs),s + (fQn),n] = 0, (2.10)

where η = (F2r H −D) is the dimensionless bottom elevation and (Qs, Qn) is a bedload

vector of modulus Φ such that fΦ is the dimensionless volumetric discharge per unitwidth of grains in the size range φ, φ+dφ. Quantities in (2.10) are made dimensionlessin the form

(η∗, L∗a) = D∗0(η, La), t∗ = t(1− p)D∗0B∗/Q0, (2.11a,b)

(Q∗s , Q∗n) = Q0(Qs, Qn), ∆ =

ρs

ρ− 1, Q0 = (∆g)1/2d

∗3/2g0 , (2.11c–e)

where p is the porosity of the mixture and g is gravity. The thickness L∗a of the activelayer is taken to be proportional to a typical coarse grain size, namely d∗σ . Recallingequation (2.8) we will assume in the following La = ks, pointing out, however, thatthis assumption is only justified in the initial phase of bar growth since it neglects theeffect of bar migration on the sediment exchange with the bed layer.

To complete the formulation we need dynamic relationships to specify bedloadintensity and the direction of trajectories that particles follow due to the combinedeffect of bottom shear stress and gravity. Following Parker (1990) we express thesediment load function Φ as

Φ = Θ3/2g G(ζ) (2.12)

where ζ is a dummy variable and

Θg =τ∗

ρ∆gd∗g0

(2.13)

denotes the dimensionless bed shear stress (Shields parameter) based on the geometricmean diameter d∗g0. Several empirical predictors of the transport capacity functionG(ζ) are available in the literature. In the following we will adopt the functionproposed by Parker (1990), which is based on field data from Oak Creek River, wherethe dummy variable ζ is given the form

ζ = ωΘg

Θr

d∗g0

d∗gghr

(d∗

d∗g

), (2.14)

and Θr = 0.0386 is a reference Shields stress, ghr is a reduced hiding function (Diplas1987; Andrews & Parker 1987; Parker 1990) which adjusts the mobility of size d∗relative to d∗g , and ω is a straining function introduced to account for the effect ofthe standard deviation of the grain size distribution on the transport rate of each sizefraction (Parker 1990).

Field data and laboratory investigations suggests the following form of ghr:

ghr = (d∗i /d∗g)−b, (2.15)


Author source of data b

Parker & Klingeman (1982) Oak Creek river 0.06Parker & Klingeman (1982) Oak Creek river 0.02a

White and Day (1982) Recirculating flume 0.19Misri, Garde & Ranga Raju (1984) Recirculating flume 0−0.08Andrews & Erman (1986) Sagehen Creek 0.13b

Wilcock (1987) Recirculating flume 0−0.03Ashworth & Ferguson (1989) Feshie and Dubhaig rivers 0.33–0.35Ashworth & Ferguson (1989) Lyngdalselva river 0.08Ferguson, Prestegaard & White river 0.12b

Ashworth (1989)Parker (1990) Oak Creek river 0.10Ashworth et al. (1992a) Sunwapta river 0.21Kuhnle (1992) Goodwin Creek river 0.19a,c

Wilcock (1992) Recirculating flume 0.27d

Wilcock & McArdell (1993) Recirculating flume 0.55e

Ferguson (1994) Roaring river 0.13b

Wathen et al. (1995) Allt Dubhaig river 0.30b

Wathen et al. (1995) Allt Dubhaig river 0.10

a Critical stress based on subsurface grain size distribution.b Critical stress inferred from maximum clast size.c Mixed gravel and sand bed.d Strongly bimodal mixture.e Very poorly sorted, bimodal mixture.Unless otherwise specified with a superscript the critical stress and, hence, the hiding factor b,

are inferred from fractional transport rates and are based on surface grain size distribution.

Table 1. Hiding exponent obtained from laboratory and field data.

with the exponent b ranging between 0 and 0.55 (see table 1). As pointed out byBuffington & Montgomery (1997) the variability of the exponent b, which is basicallya consequence of different methods adopted to estimate the critical stress, also reflectsthe role of various factors related both to sediment characteristics, and to experimentalprocedure and data analysis. In particular, significant departures from equal mobility(i.e. b = 0) are documented in the case of strongly bimodal mixtures (Kuhnle 1992;Wilcock 1992; Wilcock & McArdell 1993).

We set

(Qs, Qn) = (cos δ, sin δ)Φ (2.16)

with δ denoting the angle which the direction of bedload transport makes with thes-direction. Investigations on the direction of bedload transport of uniform sedimentsin the presence of weak spatial variations of the direction of bottom shear stress andslow spatial variations of bottom elevation (Kikkawa, Ikeda & Kitagawa 1976; Ikeda1982; Parker 1984; Olesen 1987; Sekine & Parker 1992; Talmon, Struiksma & vanMierlo 1995) suggest a prediction for δ of the form

sin δ = sin χ− r

βΘ1/2η,n (2.17)

where χ is the angle that the bottom stress vector makes with the s-axis and theparameter r ranges between 0.5 and 1.5.

Formula (2.17) can be generalized to mixtures, following the lead of Parker &Andrews (1985). Since coarser grains feel a larger ratio of transverse component


of gravitational force to downstream drag force than finer grains, it follows thatcoarser grains move preferentially downslope. The above effect can be accountedfor by introducing a suitable weighting function fhr in equation (2.17) such that thetrajectory of particles of size d∗ is defined in terms of the expression

sin δ = sin χ− r

β

[1

Θg

f−1hr

(d∗

d∗g

)]1/2

η,n, (2.18)

where we use the reduced hiding function fhr proposed by Egiazaroff (1965), asmodified by Ashida & Michiue (1972), namely

f−1hr (d∗/d∗g) =

(d∗/d∗g)(1 + 0.782 log d∗/d∗g)−2, d∗/d∗g > 0.4

0.843, d∗/d∗g < 0.4.(2.19)

Finally, in order to complete the formulation for the sediment transport we mustrequire the sidewalls to be impermeable to the sediment; hence we write

Qn = 0 (n = ± 1). (2.20)

In the next section the above formulation will be the basis for a linear stabilityanalysis of uniform configuration.

3. Linear theoryWe want to investigate the linear response of the flow field to infinitesimal per-

turbations of bottom configuration and grain size distribution. Therefore we perturbour basic state (the uniform flow and the uniformly distributed probability densityfunction of grain size) and introduce the following representation of the flow fieldand grain size distribution:

(U,V ,D,H) = (1, 0, 1, H0)

+ε (U1(s, n, t), V1(s, n, t), D1(s, n, t), H1(s, n, t)) + O(ε2), (3.1a)

f = f0(φ) + εf1(φ, s, n, t) + O(ε2), (3.1b)

with ε small parameter (strictly infinitesimal). The above expansion implies thata similar representation can be used to express the longitudinal and transversecomponents of bedload and bottom stress, namely

(Qs, Qn, τs, τn) = (Φ0, 0, C0, 0)

+ε(Qs1(φ, s, n, t), Qn1(φ, s, n, t), τs1(φ, s, n, t), τn1(φ, s, n, t)) + O(ε2), (3.2)

where Φ0 and C0 denote the bedload function and the friction coefficient of theundisturbed uniform configuration.

Furthermore we write

(φg, σ) = (φg0, σ0) + ε (φg1(s, n, t), σ1(s, n, t)) + O(ε2), (3.3a)

(η, La) = (0, La0) + ε (η1(s, n, t), La1(s, n, t)) + O(ε2) (3.3b)

where

φg0 =

∫ ∞−∞f0φ dφ, σ2

0 =

∫ ∞−∞

(φ− φg0)2f0 dφ, (3.4a,b)


φg1 =

∫ ∞−∞f1φ dφ, σ1 =

1

2σ0

∫ ∞−∞

(φ− φg0)2f1 dφ. (3.4c,d )

On substituting from (3.1a), (3.3) into the flow equations (2.3a–c) and performinglinearization at O(ε) we obtain the following differential system:

U1,s +H1,s + β(τs1 − C0D1) = 0, V1,s +H1,n + βτn1 = 0, U1,s + V1,n + D1,s = 0,

(3.5a–c)

where, using the relationships (2.6), (2.7) and (2.8), the longitudinal and transversecomponents of bottom shear stress can be expressed in the form

τs1 = C0(2U1 + D1CD) + tφφg1 + tσσ1, τn1 = C0V1, (3.6a,b)

with

tφ = −tσ = −dσ0Cd ln 2, dσ0 = dg02σ0 , dg0 = 2−φg0 , (3.7a–c)

C0 = C(1, dσ0), CD =1

C0

C,D|1,dσ0, Cd =

1

C0

C,dg |1,dσ0, (3.7d–f )

where dg0 denotes the undisturbed value of dimensionless geometric mean size whilethe subscript (1, dσ0) implies that functions are evaluated with reference to theunperturbed uniform state, namely with D = 1 and dσ = dσ0.

It is worth noting that in equation (3.6a) the effect of sorting on bed shear stressis explicitly accounted for through the dependence of τs1 on the perturbed values(φg1, σ1) of the first and the second moments of grain size distribution, whence localvariations of grain size distribution induce variations of local roughness which are feltby the fluid at a linear level as variations of longitudinal stress. The above effect hasbeen found to play a crucial role in determining bedload sheet instability (Seminara etal. 1996), while here it plays only a complementary role since the full coupling betweenperturbations of bottom topography and grain size distribution has been retained;furthermore, while in the case of bedload sheets the role of free instabilities related tosediment heterogeneity is dominant, in our case perturbations of grain size distributionare mainly driven by perturbations of bottom topography. We will see that the balancebetween the effect of bottom perturbation and that of grain size perturbation on thelongitudinal stress crucially depends on the phase lag of dg relative to bottom elevation.

The stability analysis then proceeds with the sediment balance equation at O(ε),which reads

f0(F2r H1,t − D1,t) + La0f1,t + (Φ0f1 + Qs1f0),s + (Qn1f0),n = 0. (3.8)

Linearity implies that only the unperturbed value of the thickness of the active layerLa0 = nσdσ0 appears in equation (3.8). The argument ζ of the function G (see (2.12),(2.14)) can be expanded in the form

ζ = ζ01 + ε[(2U1 + D1CD)(1 + ωT ) + ζφφg1 + ζσσ1] + O(ε2), (3.9)

with

ζ0 = ω0

Θg0

Θr

2b(φ−φg0), ζφ = (1− b) ln 2 + tφ(1 + ωT ), ζσ = tσ(1 + ωT ) + ωσ

(3.10a–c)


and

ω0 = ω(σ0, Θg0), ωσ =

[1

ω

∂ω

∂σ

]σ0,Θg0

, ωT =Θg0

Θr

[1

ω

∂ω

∂T

]σ0 ,Θg0

, (3.11a–c)

where b is the exponent of the reduced hiding function (2.15), Θg0 is Shields parameterof the basic unperturbed flow and the subscript (σ0, Θg0) indicates that quantities areevaluated with the unperturbed values of standard deviation and Shields stress.

Combining (2.16) and (2.18), and recalling the relationships (3.6) and (3.9), the O(ε)longitudinal and transverse components of bedload discharge are expressed in theform

Qs1 = Φ0(2U1 + D1CD)qη + qφφg1 + qσσ1, Qn1 = Φ0V1 − R(F2r H1,n − D1,n)

(3.12a,b)

where

qη = 32

+ (1 + ωT )Γ , qφ = 32tφ + ζφ Γ , qσ = 3

2tσ + ζσ Γ (3.13a–c)

Φ0 = Θ3/2g0 G(ζ0), Γ =

[ζ

G

dG

dζ

]ζ=ζ0

, R =r

β(Θg0fhr)1/2. (3.13d–f )

The complex coefficients qη, qφ, qσ represent the perturbations of longitudinal bed-load associated with variations of the flow field induced by bottom perturbations andby the variations of the mean and the variance of grain size distribution.

Linearity of the problem allows us to perform a normal mode analysis of theperturbations by assuming

(U1, D1, H1) = (U1(t), D1(t), H1(t)) sin ( 12πn) exp (iαs) + c.c., (3.14a)

V1 = V 1(t) cos ( 12πn) exp (iαs) + c.c., (3.14b)

(f1, φg1, σ1) = (f1(φ, t), φg1(t), σ1(t)) sin ( 12πn) exp (iαs) + c.c., (3.14c)

with α dimensionless wavenumber (scaled with 1/B∗) and c.c. denoting the complexconjugate of a complex number. For clarity only the first mode in the transversedirection is considered in (3.14a–c) and in the following. The general form of thesolution which accounts for any other transverse modes can be readily obtained.

On substituting from (3.14a,b) into (3.5) the differential system governing the flowfield is transformed into the following linear homogeneous algebraic system:

3∑j=1

[aj1U1 + aj2V 1 + aj3H1 + aj4D1 + aj5(φg1 − σ1)] = 0, (3.15)

where

a11 = iα+ 2βC0, a12 = a21 = a24 = a25 = a33 = a35 = 0, (3.16a,b)

a13 = a31 = a34 = iα, a23 = −a32 = 12π, (3.16c,d )

a14 = βC0(CD − 1), a15 = −βC0dσCd ln 2, (3.16e,f )

a22 = iα+ βC0. (3.16g)

Using (3.15) and recalling that (3.14a) implies that the bottom perturbation has a


similar structure, with η1 = F2r H1 − D1, we can readily express U1, V 1, D1, H1 in the

form

[U1, V 1, H1, D1] = (b10, b20, b30, b40)η1 + (b11, b21, b31, b41)(φg1 − σ1), (3.17)

with

b10 = − 1

iαF2r − P0

(P0 +

π2

4

1

a22

), b11 =− P1

iαF2r − P0

(iαF2

r +π2

4

1

a22

), (3.18a,b)

b20 = −π2

iα

a22

1

iαF2r − P0

, b21 = −π2

iα

a22

P1

iαF2r − P0

, (3.18c,d )

b30 =iα

iαF2r − P0

, b31 =iαP1

iαF2r − P0

, (3.18e,f )

b40 =P0

iαF2r − P0

, b41 = F2r

iαP1

iαF2r − P0

, (3.18g,h)

P0 =1

a11 − a14

[(iα)2 − a11a

223

a22

], P1 =

a15

a11 − a14

. (3.18i, j )

On substituting from (3.14c) and (3.17) into the sediment continuity equation (3.8)we obtain the following differential equation:

f0η1,t + La0f1,t = [ΓFf1 + f0(Γφφg1 + Γσσ1 + Γηη1)], (3.19)

where

Γφ = −Φ0iα(2b11 + b41CD)qη + iαqφ − 12πb21, (3.20a)

Γσ = Φ0iα(2b11 + b41CD)qη − iαqσ − 12πb21, (3.20b)

ΓF = −iαΦ0, Γη = −Φ0

iα(2b10 + b40CD)qη + 1

2π(−b20 + R 1

2π). (3.20c,d )

Finally, integrating (3.19) over the grain size and imposing the integral condition(2.2), which at order ε requires that∫ ∞

−∞f1 dφ = 0, (3.21)

we obtain the equation

η1,t =

[φg1

∫ ∞−∞f0Γφ dφ+ σ1

∫ ∞−∞f0Γσ dφ+

∫ ∞−∞f1ΓF dφ+ η1

∫ ∞−∞f0Γη dφ

]. (3.22)

Substituting η1,t from (3.22) into (3.19) we find

La0f1,t =

f0

[Γφ −

∫ ∞−∞f0Γφ dφ

]φg1 + f0

[Γσ −

∫ ∞−∞f0Γσ dφ

]σ1

+

[f1ΓF − f0

∫ ∞−∞ΓFf1 dφ

]+ f0

[Γη −

∫ ∞−∞f0Γη dφ

]η1

. (3.23)

We then obtain a system of two coupled integro-differential equations (3.22),(3.23) that constitute the main result of linear analysis. Equation (3.22) suggests thatsediment heterogeneity does affect bar instability through several contributions. In


fact, if we neglect the effect of heterogeneity of the sediment, which implies that f1

vanishes, (3.22) reduces to the usual equation which governs the linear growth of barperturbations with uniform sediments, namely

η1,t = (Ω − iω)η1, (3.24)

where Ω − iω is the complex growth rate. Moreover, bottom topography affects thespatial distribution of grain size through the fourth term appearing in (3.23). Also,

note that the unknown variables in (3.22), (3.23) are η1 and f1, since φg1 and σ1 are

related to f1 through (3.4c, d). The procedure can be readily generalized to accountfor any moment of grain size distribution.

4. Analytical solutionThe two integro-differential equations (3.22), (3.23) which govern the growth of per-

turbations of bottom elevation and grain size distribution can be solved numerically.However, an analytical treatment is possible for the case of a mixture composed ofN different discrete sizes, following a procedure similar to that adopted by Seminaraet al. (1996). We then assume the probability density function f to be represented interms of a sum of N delta Dirac functions δ(φ− φi), each centred on φi, yielding

[f0(φ), f1(φ, t)] =

N∑i=1

[f0i, f1i(t)]δ(φ− φi). (4.1)

We reformulate definitions (3.4c, d) into the discrete forms

φg1(t) =

N∑i=1

f1i(t)φi, σ1(t) =1

2σ0

N∑i=1

[f1i(t)(φi − φg0)2]. (4.2a,b)

On substituting from (4.1) and (4.2) into (3.22) we obtain

η1,t =

[N∑i=1

(f1iφi)Σ1 +1

2

N∑i=1

[f1i(φi − φg0)2]Σ2 + η1Σ3 +

N∑i=1

(f1iΓFi)

], (4.3)

where

Σ1 =

N∑i=1

(f0iΓφi

), Σ2 =

N∑i=1

(f0iΓσi) , Σ3 =

N∑i=1

(f0iΓηi

), (4.4a–c)

and the following notation is adopted:

(Γφi, Γσi, ΓFi, Γηi) =[Γφ(φi), Γσ(φi), ΓF (φi), Γη(φi)

]. (4.5)

Equation (3.23) evaluated at discrete points φi (i = 1, 2, . . . , N− 1) provides (N− 1)

equations describing the time evolution of the (N−1) perturbations f1i(t) of the grainsize distribution function

f1i,t =1

La0

f0i[Γφi − Σ1]

N∑j=1

(f1jφj) + f0i[Γσi − Σ2]1

2σ0

N∑j=1

[f1j(φj − φg0)2]

+

[f1iΓFi − f0i

N∑j=1

(ΓFjf1j)

]+ f0i[Γηi − Σ3]η1

. (4.6)


15

10

50 1 2 3

Unstable

Stable

(a)

b

15

10

5

Unstable

Stable

0 1 2 3α

(b)

b

α

Figure 2. Examples of neutral curves for alternate bar formation in the case of a bimodal mixture(dg0 = 0.01; σ0 = 2): (a) Θg0 = 0.07; (b) Θg0 = 0.10. The corresponding curves for the uniformsediment case are also reported (dotted lines).

We then obtain a linear algebraic eigenvalue problem if we write

(η1, f1i) =

N∑k=1

(e(k), ϕ(k)i ) exp (ckt), (4.7)

where ck = Ωk − iωk is the kth eigenvalue, with Ωk and ωk denoting the growthrate and the angular frequency of perturbations, while (e(k), ϕ

(k)1 , ϕ

(k)2 , . . .) is the kth

eigenvector.Note that if we set ek = 0 we reduce the system (4.3), (4.4) to the eigenvalue

problem for the bedload sheet case. The uniform sediment case is easily recovered bysetting N = 1 and leads to a dispersion relationship of the form (Colombini et al.1987)

F (Ω,ω, α, β,Θg0, dg0) = 0, (4.8)

which for given values of the unperturbed Shields stress and dimensionless geometricmean size (Θg0, dg0) allows one to define neutral conditions as those correspondingto vanishing amplification factor Ω. In the plane (α, β) this condition determines aneutral curve exhibiting a minimum which defines the critical value of width ratioβc above which bars are expected to grow and the wavelength Lc = 2παc of fastestgrowing perturbations (see figure 2).

In the case of a graded mixture made of N discrete sizes, substituting from (4.7)into (4.3), (4.6) we obtain a dispersion relationship for N eigenvalues. Though thesolution of the general case can be readily obtained, it proves to be more convenientto analyse the simplest case of a bimodal mixture (N = 2). The dispersion relationshipthen reads

(Ω − iω) = 12(ψ11 + ψ22)±

√(ψ11 − ψ22)2 + 4ψ12ψ21, (4.9)

where

ψ11 =1

La0

f01[Γφ1 − (f01Γφ1 + f02Γφ2)](φ1 − φ2) + [ΓF1 − f01(ΓF1 − ΓF2)]

+f01[Γσ1 − (f01Γσ1 + f02Γσ2)]1

2σ0

[(φ1 − φg0)2 − (φ2 − φg0)

2]

, (4.10a)

ψ12 =f01

La0[Γη1 − (f01Γη1 + f02Γη2)], (4.10b)


ψ21 = (f01Γφ1 + f02Γφ2)(φ1 − φ2) + (ΓF1 − ΓF2)

+(f01Γσ1 + f02Γσ2)1

2σ0

[(φ1 − φg0)2 − (φ2 − φg0)

2], (4.10c)

ψ22 = f01Γη1 + f02Γη2. (4.10d)

Examples of marginal curves for a bimodal mixture are reported in figure 2 fordifferent values of the Shields parameter. The comparison with the correspondingcurves of the uniform sediment case (dotted lines in the figure) suggests that sedimentheterogeneity slightly modifies the region of occurrence of bars and may lead towidening or narrowing of the unstable region depending upon the value of Shieldsstress. A detailed discussion of the effect of sorting on bar stability is deferred to thenext section.

It is worth pointing out that in the case of mixtures the dispersion relationshipdepends not only on Θg0 and dg0 but also on the standard deviation of the mixture.Also note that for a bimodal mixture composed of an equal percentage of the twofractions the quantity [(φ1−φg0)

2− (φ2−φg0)2] appearing in the coefficients ψ11 and

ψ22 vanishes.In general the dispersion relationship (4.9) provides two different eigenvalues.

The first, namely the one obtained by choosing the positive square root, recoversthe eigenvalue of the uniform sediment case as the variance of the mixture tendsto vanish, while the second degenerates into a purely imaginary solution. Whenthe perturbation of bottom amplitude is neglected the latter eigenvalue reduces tothe bedload sheet mode investigated by Seminara et al. (1996). In this case theanalysis of the limiting case of weak sorting leads to similar results in that thebedload sheet mode reduces to a sorting wave that propagates downstream withvanishing growth rate. The interested reader is referred to the above paper for furtherdetails.

5. Results and discussionAs pointed out in the Introduction, some recent experimental findings seem to

suggest that the topography of bars is significantly modified by sediment heterogeneity.A brief account of the experimental results obtained by Lanzoni et al. (1994) is givenherein. The experiments were performed in a rectangular recirculating flume, 18 mlong and 0.36 cm wide, located in the Laboratory of the Hydraulic Institute of GenoaUniversity. To investigate the effect of grain sorting on the formation of alternatebars the experimental runs were repeated, under similar hydraulic conditions, usinga uniform sediment made up of glass spheres with diameter 1.5 mm and mixtures ofglass spheres with roughly the same mean geometric grain diameter.

Figure 3(a,b) summarizes the main results of experiments with a bimodal mixturewith diameters 1 mm and 2 mm in proportion 1 : 1. According to the indexes ofbimodality introduced by Wilcock (1993) and Sambrook Smith, Nicholas & Ferguson(1997), this mixture exhibits a low degree of bimodality; hence, the hiding exponentb is likely to depart only slightly from zero (see table 1). Figure 3(a,b) shows that themajor and unambiguous effect of sediment heterogeneity in each run is the decrease ofboth height and wavelength of bars with respect to the uniform sediment case. Moreprecisely, figure 3(a) suggests that the damping effect related to sediment heterogeneityincreases as the Shields parameter decreases, leading to a maximum 50% reduction


(a) (b)3.0

2.5

2.0

1.5

1.0

0.5

00.04 0.05 0.06 0.07 0.08 0.09 0.10 0.04 0.05 0.06 0.07 0.08 0.09 0.10

40

30

20

10

0

£g0

L*b

B*H*

b

D*0

£g0

Figure 3. The experimental results of Lanzoni et al. (1994) for (a) bar height H∗b and (b) barwavelength L∗b are plotted versus Shields parameter: open circles refer to experiments with glassspheres with diameter 1.5 mm; full circles refer to experiments with bimodal mixtures of glassspheres with diameters 1 mm and 2 mm.

1.5

1.0

0.5

00.1 0.6 1.1 1.6

2.0

1.5

1.0

0.5

ó0=0

α

¿b

¿u

Figure 4. The ratio Ωb/Ωu between the growth rate of alternate bars for the bimodal case and forthe uniform case is plotted versus bar wavenumber for different values of the standard deviation ofthe mixture (dg0 = 0.001; β = 15; Θg0 = 0.08).

of bar height when Θg0 falls in the range 0.05–0.06. Moreover, the damping effectvanishes when the Shields parameter approaches the value 0.1.

The linear stability theory developed herein strictly holds for infinitesimal disturb-ances and therefore neglects finite-amplitude effects as well as the effect of the verticalpattern of sorting which is experimentally observed to form as bars approach theirequilibrium configuration (Lanzoni 1996). In spite of these restrictions the presenttheoretical results are in general agreement with the experimental observations. Thedamping effect of sorting on bar instability is clearly exhibited by theoretical resultsfor the growth rate of bottom perturbations reported in figure 4. Note that, for largeenough values of β, damping increases as the standard deviation of the mixtureincreases. Also, note that theoretical results for the growth rate include the effect ofthe variation of the overall sediment flux, which is found to increase slightly in thecase of the mixture, thus implying a small amplifying contribution to Ω.

In figure 5 the maximum growth rate is plotted versus β for some values of themean geometric diameter and different values of the unperturbed Shields stress. Thefigure allows a comparison between the results of the bimodal mixture and those


10–5

10–6

10–7

8 9 10 11 12 13 14 15 16

(a)

Max

¿

b

0.10 0.12

0.080.07

8 9 10 11 12

b76

10–4

10–5

0.07

0.08

0.10

0.12(b)

Figure 5. The theoretical values of the maximum growth rate of bars are plotted versus β for somevalues of Θg0 and σ0 = 2. The corresponding theoretical curves for the uniform sediment case arealso reported (dotted lines): (a) dg0 = 0.001; (b) dg0 = 0.01.

0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.0002

0

£g0

0.0004

0.0006

0.0008

0.0010

Max

¿

Figure 6. Theoretical values of the maximum growth rate corresponding to the experimental datareported in figure 3(a). Open and full circles refer to uniform and bimodal sediments, respectively.The calculations have been carried out by setting b = 0.2.

corresponding to the uniform sediment case (in the following these two cases will bereferred to as bimodal and uniform, respectively, and will be denoted by the subfixesb and u). In the uniform case we have used the geometric grain diameter dg0 of thebimodal mixture to compute the bedload function, while we have set the bottom grainroughness equal to nσdσ in order to keep the same friction coefficient C as the bimodalcase. Figure 5 suggests that, while sediment heterogeneity affects the threshold valueof β for bar instability only slightly, a significant damping of the growth rate of barperturbations generally occurs. Reduction of the growth rate is larger for smallervalues of Shields stress and of grain roughness. As the latter parameters increase, theeffect of sorting may also become slightly destabilizing, as indicated in figure 5(b).However, for given values of Shields stress and grain roughness, a value of β > βcalways exists such that the maximum growth rate predicted by linear theory in thebimodal case falls below the value corresponding to the uniform case. As Θg0 and dg0

decrease, this threshold value of β approaches βc.Figure 6 reports the theoretical values of the maximum growth rate corresponding

to the experimental results summarized in figure 3(a), hence allowing an indirectcomparison between theoretical predictions and experimental findings. It appearsthat the overall effect of sediment heterogeneity is adequately reproduced by the


Max

¿

8 10 12 14 16 18 20

b

10–7

10–6

Uniform

0.3

0.2

0.1 0.05 b =0

Figure 7. The maximum growth rate of bar perturbations is plotted versus β for different valuesof the exponent b of the hiding function (2.15): Θg0 = 0.08, dg0 = 0.001, σ0 = 2.

0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.001

0

£g0

0.002

0.003

0.004

0.005

x

–0.001

Figure 8. Theoretical values of the dimensionless angular frequency ω corresponding to the datareported in figure 6. Open and full circles refer to uniform and bimodal sediments, respectively. Thecalculations have been carried out by setting b = 0.2.

theoretical solution in that the damping effect is larger for low values of Θg0 andvanishes when Θg0 increases to a value approximately equal to 0.1.

It is worth pointing out that the effect of non-uniformity is enhanced at smallvalues of Θg0. In fact, sorting mainly influences bar stability through the dependenceof longitudinal and transverse bed load on grain size which is embodied in (2.12),(2.14) and (2.18). It can be readily seen that both contributions become vanishinglysmall as the Shields parameter increases: in particular, the asymptotic condition ofequal mobility is progressively met at large values of Shields stress such that thefunction G(ζ) becomes independent of the grain size.

The magnitude of the reduction of growth rate due to sediment heterogeneity isstrongly affected by the value adopted for the exponent b of the hiding function(2.15). Data reported in figures 4 and 5 are obtained with b = 0.1, which impliesthat a small (but significant) deviation from equal mobility is accounted for. Whenthe exponent b is increased within the range of observed values, the effect of sortingon the growth rate of bars becomes invariably stabilizing, as suggested by figure 7.This also indicates that the unequal response of grain sizes in a mixture to spatialvariation of bottom stress, whose effect is magnified when b is fairly large, constitutesthe main stabilizing ingredient of the present analysis. This result will be more fullydiscussed below.

A further notable feature of the linear solution is that theoretical results seem to


0.05 0.10 0.15 0.20 0.25 0.30

14

£g0 £g0

12

10

8

60.05 0.10 0.15 0.20 0.25 0.30

0.05 0.10 0.15 0.20 0.25 0.30 0.05 0.10 0.15 0.20 0.25 0.30

0.6

0.5

0.4

0.3

0.6

0.5

0.4

0.3

5

6

7

8

9

10

αc

bc

ó0 =0.2

1

2

ó0 =0.2

1

2

1ó0 =2

1

0.2

ó0 =2

0.2

(a) (b)

Figure 9. The critical values of width ratio βc and bar wavenumber αc predicted by linear theoryare plotted versus Shields parameter for some values of σ0 and dg0. The corresponding curves forthe uniform case are also reported (dotted lines): (a) dg0 = 0.001; (b) dg0 = 0.01.

reproduce the tendency of sediment heterogeneity to prevent downstream migration ofbars, which has been observed by Lisle et al. (1991) in flume experiments characterizedby shallow depth and widely graded sediments. Indeed, the theoretical values of theangular frequency, reported in figure 8 and corresponding to the experimental resultsof Lanzoni et al. (1994), support the idea that mixed-size sediments might appreciablyinhibit bar migration.

Figure 9 allows a comparison between the critical values of the wavelength andwidth ratio (i.e. the values corresponding to the minimum of a given neutral stabilitycurve) calculated both for the bimodal mixture and for the uniform sediment. Itturns out that grain sorting leads to a reduction of the wavelength selected by theinstability process, in agreement with experimental findings. Moreover, figure 9 clearlyshows that the effect of sorting on βc is fairly weak. Stabilizing effects prevail forlarger values of Θg0, while βc may be reduced due to sorting when the Shields stressapproaches its threshold value.

A closer comparison between predicted and observed bar wavelengths is difficulteven in the case of uniform sediment owing to the flatness of the marginal stabilitycurve, which implies that a relatively wide range of wavenumbers is characterized byalmost identical growth rates (see figure 4 of Colombini et al. 1987). Note, however,that in Lanzoni et al.’s (1994) experiments the relative shortening of bar length withrespect to the case of uniform sediment varies between 0.16 and 0.42, with an averagevalue of 0.32, while for the same set of data predicted values fall in the range 0.21–0.56,with an average value of 0.42.


We now attempt to provide a physical interpretation of the experimental andtheoretical results discussed above. We assume the bottom perturbation η1 to bepurely real, which implies that we set the origin of the s-axis in the section where, forn = 1, the maximum bed elevation occurs. Whence, the linear solution can be writtenin the form

U1, D1, H1, Qs1 = |u1|, |d1|, |h1|, |qS1| sin(

12πn)

exp (Ωt)

cos (αs− ωt− δu)cos (αs− ωt− δd)cos (αs− ωt− δh)cos (αs− ωt− δqs)

,

(5.1a)

V1, Qn1 = |v1|, |qn1| cos(

12πn)

exp (Ωt)

cos (αs− ωt− δv)cos (αs− ωt− δqn)

, (5.1b)

η1, f1 = e, |ϕ| sin ( 12πn) exp (Ωt)

cos (αs− ωt)cos (αs− ωt− δϕ)

, (5.1c)

where δi denotes the phase lag between the bed profile and a given variable. For a bi-modal mixture the sediment balance equation (4.3) leads to the following relationshipfor the growth rate:

Ω = 12π|qn1|e

cos (δqn)− α|qs1|e

sin (δqs)− α|qϕ|e

sin (δqϕ), (5.2)

where

qϕ = ϕ1(Φ01 − Φ02), qs1 = f01 qs1 + f02 qs2, qn1 = f01 qn1 + f02 qn2. (5.3a–c)

Note that retaining only the first two terms on the right-hand side of (5.2) we recoverthe uniform case investigated by Colombini et al. (1987). Sediment heterogeneity hasa twofold effect on Ω. In fact, the different mobility of individual grain size fractionswithin the mixture not only modifies the sediment transport capacity, namely qs1 andqn1, but also induces a longitudinal and transverse pattern of sorting. The resultingperturbation ϕ of the grain size distribution function affects in turn qs1 and qn1;moreover, it produces a further contribution, namely the third term on the right-handside of (5.2), which vanishes when the asymptotic condition of equal mobility (i.e.b = 0) holds.

Figure 10 shows a typical example of the behaviour of the various phase lags δifor the bimodal case as the wavenumber changes, for given values of the parametersΘg0, dg0, σ0, and β. Results for the uniform case are also reported. It appears thatδh, δd, δv and δqn are weakly affected by sediment non-uniformity. In particular, theflow depth is nearly in opposition to the bed profile (δd ∼ π) while the peak oftransverse velocity V1 is located about a quarter of a wavelength ahead of the peakof bottom profile, since δv is slightly larger than π/2. Furthermore, the phase lagof the transverse sediment transport δqn ranges between π/2 and π, thus implying astabilizing contribution to Ω, as indicated by (5.2). Incidentally, we may note thatthis stabilizing contribution increases with the transverse mode number; therefore, itinhibits preferentially the development of higher-order transverse modes.

The peak of the longitudinal sediment transport qs1, as well as the peaks of U1 andτs1, lag upstream of the bottom crests since π < δqs < 2π. Consequently, the secondterm on the right-hand side of (5.2) is always positive, which implies a destabilizing


(a)

(b)

2.0

1.5

1.0

0.5

00.1 0.6 1.1 1.6

dp

dU

dH

dD

dV

du1

dqs

dqu

dqn

2.0

1.5

1.0

0.5

0

dp

0.1 0.6 1.1 1.6

Bed profile

d/p

0.50 1 21.5

α

Figure 10. The phase lags of various flow and sediment properties with respect to bed profile areplotted versus bar wavenumber. Results for the uniform sediment case are reported as dotted lines.(β = 15; Θg0 = 0.08; dg0 = 0.001; σ0 = 2).

contribution. Sediment heterogeneity leads to a reduction of δqs with respect to theuniform case, thus implying a decrease (increase) of the destabilizing contributionassociated with qs1 depending on whether δqs is smaller (larger) than 3π/2.

The effect sediment non-uniformity on the amplitude of the longitudinal andtransverse components of sediment transport is illustrated in figure 11. It appearsthat according to (5.2) both |qs1| and |qn1| contribute to the reduction of the growthrate of bar perturbations with respect to the uniform case since |qs1|b/|qs1|u < 1 and|qn1|b/|qn1|u > 1.

The effect on |qn1| can be easily understood through direct inspection of (3.12b)which suggests that sediment non-uniformity enters into the problem through thedependence of the transport function Φ and of the coefficient R on the grain size. Ifwe keep only the contribution related to transverse bedload, equation (5.2) reducesto the expression

Ω = 14π2Θ

3/2g0 f01G(φ1)R(φ1) + f02G(φ2)R(φ2). (5.4)

Therefore, the stabilizing effect associated with Qn1 increases with respect to theuniform case according to the ratio

f01G(φ1)[fhr(φ1)]−1 + f02G(φ2)[fhr(φ2)]

−1

G(φg0),


1.2

0.8

0.4

00.1 0.6 1.1 1.6

α

rqnrb/rqnru

rqsrb/rqsru

rqurb/rqsru

Figure 11. The ratio between the amplitudes of the perturbations of longitudinal andtransverse components of sediment transport calculated for the bimodal case and for theuniform case is plotted versus bar wavenumber. The ratio |qϕ|b/|qs|u is also reported.(β = 15;Θg0 = 0.08; dg0 = 0.001; σ0 = 2).

which is always greater than 1 and increases as the variance of the mixture increases.Regarding the reduction of |qs1| with respect to the uniform case reported in figure

11, it is not easy to analyse separately the role played by the various terms contributingto Qs1 in (3.12a). However, we may observe that qη , namely the contribution associatedwith the variation of the transport function G with respect to the Shields parameter,decreases with respect to the uniform case, the reduction being more appreciable asthe Shields parameter decreases.

A direct effect of grain sorting on Ω is embodied in the third term on the right-handside of (5.2); it appears that this contribution always induces damping of the growthrate of bar perturbations since π/2 < δqϕ < π. This stabilizing effect is essentiallyrelated to the progressive coarsening which takes place along the upstream face ofbars, namely near the peak of bottom shear stress, as a consequence of the selectivetransport of different grain size fractions. In fact, the derivative of the sediment loadfunction Φ with respect to the shear stress τ is such that the ratio

Φ,τ|diΦ,τ|dg =

G(φi)

G(φg)

32

+ Γ (φi)32

+ Γ (φg)(5.5)

is greater or smaller than 1, depending on whether di is finer or coarser than dg ,respectively. As a consequence, different grain sizes display a different response tothe increasing values of bed shear stress which occur along the rising part of thebottom profile, leading to coarsening of the bed composition since finer particlesare more easily transported. Indeed, figure 10 shows that the phase lag δϕ1 of theperturbation of grain size distribution ranges between 3π/2 and 2π, which impliesthat the coarser fraction prevails upstream of the bar crest. The above findings agreewith the experimental observations of Lisle et al. (1991), Lisle & Madej (1992) andLanzoni (1996).

The above tendency is counteracted by gravitational effects which tend to pullcoarser particles downward selectively (see Parker & Andrews 1985). The experimentalresults seem to suggest that the former mechanism, namely the unequal response ofgrain sizes to bottom stress, prevails in the case of bars since spatial variationsof bottom topography are relatively slow. It may turn out that gravity becomesthe dominant mechanism which controls the location of grain sizes in the case of


1

0

–1

–20 0.3 0.6 1.5

α0.9 1.2

2

3

¿

(×10–6)

Figure 12. The theoretical predictions of the growth rate of alternate bars for a bimodal mixture(solid line) are compared with the values of Ω obtained in the uniform sediment case (dottedline) and with the values obtained in the bimodal case neglecting the effect of sorting on thelongitudinal (· · · + · · · , ϕ = 0, b = 0) and transverse (· · · • · · · , fhr = 0) component of bedload:β = 15, Θg0 = 0.08, dg0 = 0.001, σ0 = 2.

bedforms of smaller spatial scale, such as dunes, or when finite-amplitude effects aremore prominent.

A further consequence of the sorting pattern induced by bar topography is that itimplies a reduction of longitudinal sediment transport with respect to the uniform casein the region where less mobile coarser particles preferentially accumulate, namelyalong the upstream face of bars. Therefore, a positive contribution to qs,s is generatedin the neighbourhood of a bar crest, which implies a negative contribution to bottomdevelopment, according to equation (3.8).

Regarding the effects of grain sorting on the migration of bars, theoretical resultssupport the idea, reported by Lisle et al. (1991), that bar coarsening is mainlyresponsible for the observed stabilization of bars. In fact, the effect of sedimentheterogeneity on the wave speed of bars is felt essentially through the dependence ofω on the perturbation of the grain size distribution ϕ. From (4.3) and (5.1) we obtain

ω ∝ cos (δqϕ). (5.6)

Hence, bar coarsening, which implies that δqϕ falls in the range π/2–π as discussedabove, leads to a negative contribution to the wave speed.

An example of how the various mechanisms affect the growth rate of bar pertur-bations is given in figure 12. From the preceeding discussion it clearly emerges that,within a fairly wide range of values of the dimensionless control parameters (Θg0, dg0

and β), various contributions related to sediment heterogeneity act simultaneouslyto produce an overall stabilizing effect on bottom development, which leads to anon-negligible reduction of the depths of scour and deposition associated with barformation, and a damping of migration speed. This is the main practical implicationof the present work, which mostly applies to gravel bed rivers. The theoretical modelpresented herein seems to reproduce adequately the main features of bar develop-ment related to the heterogeneous character of sediments which have been observedexperimentally, namely the overall stabilizing effect on bar height, the shortening ofbar wavelengths, the decrease of migration speed, as well as the pattern of sortinginduced by bottom development.

However, predicting the equilibrium bar morphology would require that the as-sumption of small perturbations be removed. To this end the development of a sound


model for vertical sorting is required as the first step to incorporate in the analysisfinite-amplitude effects on the sorting process. Furthermore, the introduction of sus-pended load effects may allow the extension of the theory to the case when the grainsize distribution includes the sand range. In this case it is possible that the effect onbed stability of sediment heterogeneity may become even more pronounced due tothe strong selective character of suspended load.

This work has been financially supported by the Italian Ministry of ScientificResearch (MPI 40% National Project ‘Trasporto di sedimenti ed evoluzione morfo-logica di corsi d’acqua, estuari e lagune alle diverse scale temporali’ and COFIN ‘97‘Morfodinamica Fluviale e Costiera’). It is a pleasure to thank P. J. Ashworth for histhorough and constructive review.

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